If R is an integral domain with unit element, prove that any unit in R[x] must already be a unit in R.

Modern Algebra Ring Theory (XL)
Polynomial Rings over Commutative rings Integral Domain
Unit Element
Unit of a Commutative Ring

If R is an integral domain with unit element, prove that any unit in R[x] must already be a unit in R.

Solution Summary

This solution is comprised of a detailed explanation of Polynomial Rings over Commutative Rings.It contains step-by-step explanation that if R is an integral domain with unit element, then any unit in R[x] must already be a unit in R.

... of the properties of polynomial rings over commutative rings or integral domains. ... A commutative ring R is said to be an integral domain if R has no zero ...

... Since R is an integral domain, thus R is a field. ... c) We can borrow the same proof in (b). If a in ... Since R has identity 1, then we can find b in R, such that f(b ...

... A commutative ring R is said to be an integral domain if R has ... of a polynomial Let f ( x ) ∈ F [x ] . If f ( x... a0 , a1 , a2 ,..., an are all in R and where n ...

... a principal ideal domain, Theorem 1. Let D be an integral domain. If the polynomial ring then D is a field. ... Then the ideal = ( , ) generated by r and x must be ...

...If RCF, where F is a field, show that F has a sub-field K such that RCK and K is isomorphic to Q. Proof: R is an integral domain, it means R is a ...

... another way, recall that if R is an integral domain, and S ... is the field of fractions of R; and if F is ...R, then the subfield of F generated by R' is isomorphic ...

... Group Policy objects (GPOs) based on server role is an integral piece of the ...R EFERENCES. ... for SRS Section 3 Organized by functional hierarchy (Good if you have ...